Same Odds, Different Deals: How Risk Framing Shapes AI Collusion

Collusion-deterrence evaluations usually state an enforcement risk to an AI agent and count how often it proposes a prohibited deal. They implicitly treat the agent as responding to the odds alone. We test whether the same odds, described differently, produce the same deals. In a scored verification task with executed lotteries, we ran three preregistered blocks with DeepSeek-V3.2, GPT-4.1-mini and Llama-3.3-70B (26,820 scored decisions). Holding probability and penalty fixed, we varied only the description. Proposal rates moved by up to 19 points when a partner rather than an audit might report, by up to 38 points when "10%" became "1 report in 10", and by up to 71 points when a single past report was recent rather than old. Directions differed by model, and DeepSeek still stated the correct probabilities when asked separately. A truthful one-line risk summary closed most of the gap in one model but pushed another to 98% proposals. Deterred agents sent fewer proposals but did not make them harder to detect: a fixed LLM reporter's report probability changed by at most 1.4 points. Expected simulated misconduct fell with sending, though proportionally less than expected reports, an exploratory gap that does not survive multiplicity correction. Proposals also rarely became completed deals: live partners accepted 0 of 180 drafts and carried out 1 of 60 three-round conversations. Deterrence measured under one risk description is therefore fragile. Evaluators should test matched descriptions and check proposal counts against outcomes.